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# HW3A - through the two observations can be very di/erent...

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Econ 444 Elementary Econometrics (Fall 2007) ANSWERS TO HOMEWORK EXERCISE: 3 (a) V AR ( ° i ) = E ( ° 2 ) . ° 2 is 3 2 with probability 0.5 and ( ° 3) 2 with prob- ability 0.5. Hence V AR ( ° i ) = 0 : 5 ± 9 + 0 : 5 ± 9 = 9 : (b) From (a), ± 2 = 9 . The sample mean of X is ° X = (32+34+42) = 3 = 36 . Hence P 3 i =1 ( X ° ° X ) 2 = ( ° 4) 2 +( ° 2) 2 +6 2 = 16+4+36 = 56 . Using Equation (5), V AR ( ^ ² 1 ) = 9 = 56 = 0 : 161 . (c) Using Equation (8), the variance is 2 ± 9 (42 ° 34) 2 = 0 : 281 : (1) (d) Using Equation (8), the variance is 2 ± 9 (34 ° 32) 2 = 4 : 5 : (2) (e) Using Equation (8), the variance is 2 ± 9 (42 ° 32) 2 = 0 : 18 : (3) (f) Comparing the results for (c)-(e), the two-point estimator with i=1 and j=2 in (d) has the maximum variance. The two-point estimator with i=1 and j=3 in (e) has the minimum variance. (g) Among the two-point estimators, the variance should increase as the distance between X j and X i decreases. This is because the slope that goes

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Unformatted text preview: through the two observations can be very di/erent from the true slope co-e¢ cient when X j and X i are close. Because X 1 and X 2 are closer than X 2 and X 3 in this example, the maximum variance results when we choose i = 1 and j = 2 . Because the distance between X j and X i is the largest when we choose X 1 and X 3 , the minimum variance results when we choose i = 1 and j = 3 . (h) The OLS estimator in (b) has the minimum variance. 1 (i) Yes. The Gauss-Markov Theorem states that the OLS estimator is the BLUE. Because the two-point estimators are linear unbiased estimators, the variance of the OLS estimator must be smaller than or equal to the variance of the any two-point estimator. 2...
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HW3A - through the two observations can be very di/erent...

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